{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:DHWBISD5XZH5CMAANRJADQFYX7","short_pith_number":"pith:DHWBISD5","schema_version":"1.0","canonical_sha256":"19ec14487dbe4fd130006c5201c0b8bfcb18ea95b343cd61c025c6da21e8a558","source":{"kind":"arxiv","id":"1210.2653","version":1},"attestation_state":"computed","paper":{"title":"Compactness and Bubbles Analysis for 1/2-harmonic Maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Francesca Da Lio","submitted_at":"2012-10-09T16:09:32Z","abstract_excerpt":"In this paper we study compactness and quantization properties of sequences of 1/2-harmonic maps $u_k\\colon\\R\\to {\\cal{S}}^{m-1}$ such that $|u_k|_{\\dot H^{1/2}(\\R,{\\cal{S}}^{m-1})}\\le C.$ More precisely we show that there exist a weak 1/2-harmonic map $u_\\infty\\colon\\R\\to {\\cal{S}}^{m-1}$, a possible empty set ${a_1,...,a_\\ell}$ in $\\R$ such that up to subsequences $$(|(-\\Delta)^{1/4}u_k|^2 \\rightharpoonup |(-\\Delta)^{1/4}u_{\\infty}|^2)dx+\\sum_{i=1}^{\\ell}\\lambda_i \\delta_{a_i}, in Radon measure,$$ as $k\\to +\\infty$, with $\\lambda_i\\ge 0.$\n  The convergence of $u_k$ to $u_\\infty$ is strong in"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1210.2653","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-10-09T16:09:32Z","cross_cats_sorted":[],"title_canon_sha256":"5023624bb91b9fe40e856d2e205bc29bad995057a7caef1644efd36e574ccfcc","abstract_canon_sha256":"6e1a8c82b58a15db53f3aedc58ef42b6d467fa9077c2a5150dca28131ca2265e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:43:41.029694Z","signature_b64":"gZo/3xjyOaZXjvUA3CeWbrD/GC05xZZOjcXhOPLXzBP6kkfyMfs9Q2oC2V/yaU+WlLkFBbGuVh7WF6Yk3ojCDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"19ec14487dbe4fd130006c5201c0b8bfcb18ea95b343cd61c025c6da21e8a558","last_reissued_at":"2026-05-18T03:43:41.029120Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:43:41.029120Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Compactness and Bubbles Analysis for 1/2-harmonic Maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Francesca Da Lio","submitted_at":"2012-10-09T16:09:32Z","abstract_excerpt":"In this paper we study compactness and quantization properties of sequences of 1/2-harmonic maps $u_k\\colon\\R\\to {\\cal{S}}^{m-1}$ such that $|u_k|_{\\dot H^{1/2}(\\R,{\\cal{S}}^{m-1})}\\le C.$ More precisely we show that there exist a weak 1/2-harmonic map $u_\\infty\\colon\\R\\to {\\cal{S}}^{m-1}$, a possible empty set ${a_1,...,a_\\ell}$ in $\\R$ such that up to subsequences $$(|(-\\Delta)^{1/4}u_k|^2 \\rightharpoonup |(-\\Delta)^{1/4}u_{\\infty}|^2)dx+\\sum_{i=1}^{\\ell}\\lambda_i \\delta_{a_i}, in Radon measure,$$ as $k\\to +\\infty$, with $\\lambda_i\\ge 0.$\n  The convergence of $u_k$ to $u_\\infty$ is strong in"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.2653","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1210.2653","created_at":"2026-05-18T03:43:41.029219+00:00"},{"alias_kind":"arxiv_version","alias_value":"1210.2653v1","created_at":"2026-05-18T03:43:41.029219+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.2653","created_at":"2026-05-18T03:43:41.029219+00:00"},{"alias_kind":"pith_short_12","alias_value":"DHWBISD5XZH5","created_at":"2026-05-18T12:27:04.183437+00:00"},{"alias_kind":"pith_short_16","alias_value":"DHWBISD5XZH5CMAA","created_at":"2026-05-18T12:27:04.183437+00:00"},{"alias_kind":"pith_short_8","alias_value":"DHWBISD5","created_at":"2026-05-18T12:27:04.183437+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/DHWBISD5XZH5CMAANRJADQFYX7","json":"https://pith.science/pith/DHWBISD5XZH5CMAANRJADQFYX7.json","graph_json":"https://pith.science/api/pith-number/DHWBISD5XZH5CMAANRJADQFYX7/graph.json","events_json":"https://pith.science/api/pith-number/DHWBISD5XZH5CMAANRJADQFYX7/events.json","paper":"https://pith.science/paper/DHWBISD5"},"agent_actions":{"view_html":"https://pith.science/pith/DHWBISD5XZH5CMAANRJADQFYX7","download_json":"https://pith.science/pith/DHWBISD5XZH5CMAANRJADQFYX7.json","view_paper":"https://pith.science/paper/DHWBISD5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1210.2653&json=true","fetch_graph":"https://pith.science/api/pith-number/DHWBISD5XZH5CMAANRJADQFYX7/graph.json","fetch_events":"https://pith.science/api/pith-number/DHWBISD5XZH5CMAANRJADQFYX7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DHWBISD5XZH5CMAANRJADQFYX7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DHWBISD5XZH5CMAANRJADQFYX7/action/storage_attestation","attest_author":"https://pith.science/pith/DHWBISD5XZH5CMAANRJADQFYX7/action/author_attestation","sign_citation":"https://pith.science/pith/DHWBISD5XZH5CMAANRJADQFYX7/action/citation_signature","submit_replication":"https://pith.science/pith/DHWBISD5XZH5CMAANRJADQFYX7/action/replication_record"}},"created_at":"2026-05-18T03:43:41.029219+00:00","updated_at":"2026-05-18T03:43:41.029219+00:00"}